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Theorem tanatan 23376
Description: The arctangent function is an inverse to  tan. (Contributed by Mario Carneiro, 2-Apr-2015.)
Assertion
Ref Expression
tanatan  |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A )
)  =  A )

Proof of Theorem tanatan
StepHypRef Expression
1 atancl 23338 . . 3  |-  ( A  e.  dom arctan  ->  (arctan `  A )  e.  CC )
2 2efiatan 23375 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 ) )
32oveq1d 6311 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  +  1 ) )
4 2mulicn 10783 . . . . . . . 8  |-  ( 2  x.  _i )  e.  CC
54a1i 11 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  e.  CC )
6 atandm 23333 . . . . . . . . 9  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
76simp1bi 1011 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  A  e.  CC )
8 ax-icn 9568 . . . . . . . 8  |-  _i  e.  CC
9 addcl 9591 . . . . . . . 8  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  +  _i )  e.  CC )
107, 8, 9sylancl 662 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  e.  CC )
11 subneg 9887 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  -  -u _i )  =  ( A  +  _i ) )
127, 8, 11sylancl 662 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =  ( A  +  _i ) )
136simp2bi 1012 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  A  =/=  -u _i )
148negcli 9906 . . . . . . . . . 10  |-  -u _i  e.  CC
15 subeq0 9864 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =  0  <->  A  =  -u _i ) )
1615necon3bid 2715 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
177, 14, 16sylancl 662 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
1813, 17mpbird 232 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =/=  0 )
1912, 18eqnetrrd 2751 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  =/=  0 )
205, 10, 19divcld 10341 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  /  ( A  +  _i ) )  e.  CC )
21 ax-1cn 9567 . . . . . 6  |-  1  e.  CC
22 npcan 9848 . . . . . 6  |-  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 )  +  1 )  =  ( ( 2  x.  _i )  / 
( A  +  _i ) ) )
2320, 21, 22sylancl 662 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  +  1 )  =  ( ( 2  x.  _i )  /  ( A  +  _i )
) )
243, 23eqtrd 2498 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =  ( ( 2  x.  _i )  /  ( A  +  _i ) ) )
25 2muline0 10784 . . . . . 6  |-  ( 2  x.  _i )  =/=  0
2625a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  =/=  0 )
275, 10, 26, 19divne0d 10357 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  /  ( A  +  _i ) )  =/=  0
)
2824, 27eqnetrd 2750 . . 3  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =/=  0 )
29 tanval3 13881 . . 3  |-  ( ( (arctan `  A )  e.  CC  /\  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =/=  0 )  ->  ( tan `  (arctan `  A ) )  =  ( ( ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) ) )
301, 28, 29syl2anc 661 . 2  |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A )
)  =  ( ( ( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) ) )
312oveq1d 6311 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  -  1 )  =  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  -  1 ) )
3221a1i 11 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  1  e.  CC )
3320, 32, 32subsub4d 9981 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  -  1 )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  (
1  +  1 ) ) )
34 df-2 10615 . . . . . . . 8  |-  2  =  ( 1  +  1 )
3534oveq2i 6307 . . . . . . 7  |-  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  (
1  +  1 ) )
3633, 35syl6eqr 2516 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  -  1 )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  2 ) )
3731, 36eqtrd 2498 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  -  1 )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 ) )
38 2cn 10627 . . . . . . . 8  |-  2  e.  CC
39 mulcl 9593 . . . . . . . 8  |-  ( ( 2  e.  CC  /\  ( A  +  _i )  e.  CC )  ->  ( 2  x.  ( A  +  _i )
)  e.  CC )
4038, 10, 39sylancr 663 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( A  +  _i ) )  e.  CC )
415, 40, 10, 19divsubdird 10380 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  -  ( 2  x.  ( A  +  _i ) ) )  / 
( A  +  _i ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  ( ( 2  x.  ( A  +  _i ) )  /  ( A  +  _i ) ) ) )
42 mulneg12 10016 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( -u 2  x.  A )  =  ( 2  x.  -u A
) )
4338, 7, 42sylancr 663 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( -u
2  x.  A )  =  ( 2  x.  -u A ) )
44 negsub 9886 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  +  -u A )  =  ( _i  -  A ) )
458, 7, 44sylancr 663 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( _i  +  -u A )  =  ( _i  -  A
) )
4645oveq1d 6311 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( _i  +  -u A
)  -  _i )  =  ( ( _i 
-  A )  -  _i ) )
477negcld 9937 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  -u A  e.  CC )
48 pncan2 9846 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  -u A  e.  CC )  ->  ( ( _i  +  -u A )  -  _i )  =  -u A
)
498, 47, 48sylancr 663 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( _i  +  -u A
)  -  _i )  =  -u A )
508a1i 11 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
5150, 7, 50subsub4d 9981 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( _i  -  A )  -  _i )  =  ( _i  -  ( A  +  _i )
) )
5246, 49, 513eqtr3rd 2507 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( _i 
-  ( A  +  _i ) )  =  -u A )
5352oveq2d 6312 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  -  ( A  +  _i ) ) )  =  ( 2  x.  -u A
) )
5438a1i 11 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  2  e.  CC )
5554, 50, 10subdid 10033 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  -  ( A  +  _i ) ) )  =  ( ( 2  x.  _i )  -  (
2  x.  ( A  +  _i ) ) ) )
5643, 53, 553eqtr2rd 2505 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  -  ( 2  x.  ( A  +  _i ) ) )  =  ( -u 2  x.  A ) )
5756oveq1d 6311 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  -  ( 2  x.  ( A  +  _i ) ) )  / 
( A  +  _i ) )  =  ( ( -u 2  x.  A )  /  ( A  +  _i )
) )
5854, 10, 19divcan4d 10347 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( A  +  _i ) )  /  ( A  +  _i ) )  =  2 )
5958oveq2d 6312 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  (
( 2  x.  ( A  +  _i )
)  /  ( A  +  _i ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 ) )
6041, 57, 593eqtr3d 2506 . . . . 5  |-  ( A  e.  dom arctan  ->  ( (
-u 2  x.  A
)  /  ( A  +  _i ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 ) )
6137, 60eqtr4d 2501 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  -  1 )  =  ( (
-u 2  x.  A
)  /  ( A  +  _i ) ) )
6224oveq2d 6312 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  +  1 ) )  =  ( _i  x.  ( ( 2  x.  _i )  /  ( A  +  _i )
) ) )
638, 38, 8mul12i 9792 . . . . . . . 8  |-  ( _i  x.  ( 2  x.  _i ) )  =  ( 2  x.  (
_i  x.  _i )
)
64 ixi 10199 . . . . . . . . 9  |-  ( _i  x.  _i )  = 
-u 1
6564oveq2i 6307 . . . . . . . 8  |-  ( 2  x.  ( _i  x.  _i ) )  =  ( 2  x.  -u 1
)
6621negcli 9906 . . . . . . . . 9  |-  -u 1  e.  CC
6738mulm1i 10022 . . . . . . . . 9  |-  ( -u
1  x.  2 )  =  -u 2
6866, 38, 67mulcomli 9620 . . . . . . . 8  |-  ( 2  x.  -u 1 )  = 
-u 2
6963, 65, 683eqtri 2490 . . . . . . 7  |-  ( _i  x.  ( 2  x.  _i ) )  = 
-u 2
7069oveq1i 6306 . . . . . 6  |-  ( ( _i  x.  ( 2  x.  _i ) )  /  ( A  +  _i ) )  =  (
-u 2  /  ( A  +  _i )
)
7150, 5, 10, 19divassd 10376 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  ( 2  x.  _i ) )  /  ( A  +  _i ) )  =  ( _i  x.  ( ( 2  x.  _i )  /  ( A  +  _i ) ) ) )
7270, 71syl5eqr 2512 . . . . 5  |-  ( A  e.  dom arctan  ->  ( -u
2  /  ( A  +  _i ) )  =  ( _i  x.  ( ( 2  x.  _i )  /  ( A  +  _i )
) ) )
7362, 72eqtr4d 2501 . . . 4  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  +  1 ) )  =  ( -u 2  /  ( A  +  _i ) ) )
7461, 73oveq12d 6314 . . 3  |-  ( A  e.  dom arctan  ->  ( ( ( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) )  =  ( ( ( -u 2  x.  A )  /  ( A  +  _i )
)  /  ( -u
2  /  ( A  +  _i ) ) ) )
7538negcli 9906 . . . . . 6  |-  -u 2  e.  CC
76 mulcl 9593 . . . . . 6  |-  ( (
-u 2  e.  CC  /\  A  e.  CC )  ->  ( -u 2  x.  A )  e.  CC )
7775, 7, 76sylancr 663 . . . . 5  |-  ( A  e.  dom arctan  ->  ( -u
2  x.  A )  e.  CC )
7875a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  -u 2  e.  CC )
79 2ne0 10649 . . . . . . 7  |-  2  =/=  0
8038, 79negne0i 9913 . . . . . 6  |-  -u 2  =/=  0
8180a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  -u 2  =/=  0 )
8277, 78, 10, 81, 19divcan7d 10369 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( ( -u 2  x.  A )  /  ( A  +  _i )
)  /  ( -u
2  /  ( A  +  _i ) ) )  =  ( (
-u 2  x.  A
)  /  -u 2
) )
837, 78, 81divcan3d 10346 . . . 4  |-  ( A  e.  dom arctan  ->  ( (
-u 2  x.  A
)  /  -u 2
)  =  A )
8482, 83eqtrd 2498 . . 3  |-  ( A  e.  dom arctan  ->  ( ( ( -u 2  x.  A )  /  ( A  +  _i )
)  /  ( -u
2  /  ( A  +  _i ) ) )  =  A )
8574, 84eqtrd 2498 . 2  |-  ( A  e.  dom arctan  ->  ( ( ( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) )  =  A )
8630, 85eqtrd 2498 1  |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A )
)  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   dom cdm 5008   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510   _ici 9511    + caddc 9512    x. cmul 9514    - cmin 9824   -ucneg 9825    / cdiv 10227   2c2 10606   expce 13809   tanctan 13813  arctancatan 23321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-fac 12357  df-bc 12384  df-hash 12409  df-shft 12912  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-ef 13815  df-sin 13817  df-cos 13818  df-tan 13819  df-pi 13820  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-limc 22396  df-dv 22397  df-log 23070  df-atan 23324
This theorem is referenced by:  atantanb  23381  atanord  23384
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